(2) By [1]}, two Artin \(R\) -algebras \(A\)  and \(B\) 
are said to be derived equivalent if their derived categories
\({\rm D^b}(A)\)  and \({\rm D^b}(B)\)  are equivalent as triangulated categories. By
Theorem 2.1, Artin algebras \(A\)  and \(B\)  are derived equivalent if
and only if \(B\)  is isomorphic to the endomorphism algebra of a
tilting complex \(T^{\bullet }\) . If \(T^{\bullet }\)  is a tilting complex for
\(A\) , then there is an equivalence \(F: {\rm D^b}(A)\rightarrow {\rm D^b}(B)\)  that sends
\(T^{\bullet }\)  to \(B\) . On the other hand, for each derived equivalence
\(F: {\rm D^b}(A)\rightarrow {\rm D^b}(B)\) , there is an associated tilting complex
\(T^{\bullet }\)  for \(A\)  such that \(F(T^{\bullet })\)  is isomorphic to \(B\)  in
\({\rm D^b}(B)\) .
